55 problems

Let $P_0,\ldots,P_n$ be homogeneous polynomials in $cmathbb{C}[x_0,\ldots,x_n]$ with no common zeroes in $cmathbb{P}^n$, and set … For a general linear form $cell$ in $A_1$, consid…

Migliore–Miro-Roig–Nagel's conjecture. When $n\geq 9$ is odd, this algebra should fail the WLP.

Let $A$ be an artinian Gorenstein algebra of socle degree $d\geq 3$ whose defining ideal is generated by quadrics. The weak Lefschetz property means that multiplication by a genera…

Let $P\subset\mathbb{R}^d$ be a lattice polytope. It has the integer decomposition property (IDP) if every lattice point of $\operatorname{cone}(P)$ is a nonnegative integral combi…

Let $n$ be a positive integer, let $\mathbb{K}$ be a field of characteristic zero, and let $f\in\mathbb{K}[x_1,\dots,x_n]$ be a homogeneous polynomial. Set…

Let $\sigma \\: \Gamma \to 2^V$ be a vertex-induced triangulation of a simplex, and let $k$ be a field of characteristic $0$. A generic special linear system of parameters is the s…

Bjorner--Swartz conjecture. Every doubly Cohen--Macaulay simplicial complex has the dual WLP.

Let $\Delta$ be a balanced $d$-sphere with coloring $\rho: V \to [d+1]$, let $R$ be the polynomial ring whose variables $x_v$ have color $\rho(v)$, and let … Set…

Let $\Delta$ be a $(d-1)$-dimensional orientable homology manifold over $k$, let $k[\Delta]$ be its face ring, and let $\Theta\subseteq k[\Delta]$ be a linear system of parameters.…

Let $S_d(U)$ be a squeezed sphere, or an S-squeezed sphere, constructed from a shifted order ideal $U$. A simplicial sphere has the strong Lefschetz property when its Lefschetz map…

Let $I \subset R=k[x_1,\dots,x_n]$ be generated by $r$ general forms, and let $c$ be the maximal socle degree of $R/I$, meaning the last degree in which $R/I$ is nonzero. A non-Kos…

Let $C\subset\mathbb P^3$ be the union of $r$ general lines. The strong Lefschetz property conjecture. Then $M(C)$ has SLP. Here $M(C)$ denotes the Hartshorne–Rao module of $C$, an…

Ragunathan–Van Tuyl's conjecture. Fix an integer $k\geq 3$. If $k$ is odd, then there exists an $m$ such that $A(n,k)$ fails to have the WLP for every $n\geq m$. If $k$ is even, th…

Recursive description of the integers $c_j$. If $b-j\leq rq$, then $c_j(a,b)=rq$. Otherwise,

For a graded Artinian algebra $A$ over a field $k$, the strong Lefschetz property means that there exists $z\in A_1$ such that the multiplication map … has maximal rank for every…

Cruz–Iarrobino's socle-dimension conjecture. The dimension of the socle equals $2^{n/2}$ when $n$ is even, and $1$ when $n$ is odd.

Let $k$ be a field of arbitrary characteristic and let $\Delta$ be a connected oriented simplicial pseudomanifold over $k$ of dimension $d-1$ with vertex set $V$. Let $\ell$ be the…

Let $k$ be a field and let $\Delta$ be a connected oriented simplicial pseudomanifold over $k$. The constructions and results above, including the odd-multiplicity Hodge–Riemann th…

Let $cmathcal{E}$ be a general semistable rank-$2$ vector bundle on $cmathbb{P}^2$ with even first Chern class, let $M=cmathop{H}^1_*(cmathbb{P}^2,cmathcal{E})$, and let…

Almkvist–Lefschetz strengthening. (i) For each $m$, the algebra $A(m,n)$ satisfies the SLP for all $n\geq 11$. (ii) Moreover, if $m$ is even, then $A(m,n)$ satisfies the SLP for al…

Let $\pmb{\lambda}$ and $k$ be as in the generalized Kronecker-coefficient conjecture, and let $\omega\in\bigwedge^k V$ be the Cayley form. Define $L(v)=\omega\wedge v$. The proper…

Codimension-three Artinian Gorenstein WLP conjecture. Every such algebra $A$ has the WLP. The source notes that codimension-three standard-graded complete intersections have the WL…

Let $R=k[x_1,\dots,x_n]$ where $k$ is a field of characteristic zero. Fix $n,d\geq 3$ and let $\mu$ lie in the interval $\left[n,\binom{n+d-1}{d}\right]$. A monomial ideal $I$ mini…

WLP conjecture. Every Artinian Gorenstein quotient of $k[x,y,z]$ has the WLP.

Let $A$ be an Artin Gorenstein algebra with Hilbert vector $H(A)=(1,4,7,7,4,1)$, and let the Betti tables below refer to the graded Betti tables of such algebras. Nine-table and st…